Problem. $(X,\mathcal{B}, \mu)$ be a finite measure space where $\mathcal{B}$ is the $\sigma$-algebra and $\mu$ is the measure. Let $f$ be measurable on $X$. Let $A_n:=\{x\in X: n-1\leq |f(x)|<n\}$ for $n=1,2,...$ Show that $f\in \mathcal{L}^p(\mu)$ for $1\leq p<\infty$ if and only if $\sum_{n=1}^{\infty} n^p \mu(A_n)<\infty$
So my idea is to write $X=\bigcup_{n=1}^{\infty} A_n$, as these are disjoint then we can use the identity(taken in the power of $p$) $$ \int_X f(x)d\mu=\sum_{n=1}^{\infty}\int_{A_n}f d\mu$$ but this gives only part of the condition. I'm not sure where the $n^p$ factor in the sum comes from.